Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the number of ways of partitioning a positive number $k\leq mn$ using non-increasing parts such that the number of parts can be at most $n$ and the value of each part can be at most $m$?

share|improve this question

1 Answer 1

You are asking for the coefficients of the Gaussian or $q$-binomial coefficient. If $$ [n] := \frac{q^n-1}{q-1} $$ and $$ [n+1]! := [n+1]\cdot [n]!, \quad [0]!=1 $$ then you want the coefficient of $q^k$ in the $q$-binomial coefficient $$ \frac{[m+n]!}{[m]![n]!}. $$ There is no explicit formula. Googling on Gaussian binomial should lead to a proof.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.